Some advances in the theory of voting systems based on experimental algorithms ⋆ Josep Freixas 1 and Xavier Molinero 2 Escola Polit` ecnica Superior d’Enginyeria de Manresa E-08242 Manresa, Spain. Universitat Polit` ecnica de Catalunya. 1 Dept. de Matem`atica Aplicada 3. E-mail: josep.freixas@upc.edu. 2 Dept. de Llenguatges i Sistemes Inform`atics. E-mail: molinero@lsi.upc.edu. Abstract. In voting systems, game theory, switching functions, thresh- old logic, hypergraphs or coherent structures there is an important prob- lem that consists in determining the weightedness of a voting system by means of trades among voters in sets of coalitions. The fundamental theorem by Taylor and Zwicker [8] establishes the equivalence between weighted voting games and k-trade robust games for each positive in- teger k. Moreover, they also construct, in [9], a succession of games G k based on magic squares which are (k - 1)-trade robust but not k-trade robust, each one of these games G k has k 2 players. The goal of this paper is to provide improvements by means of differ- ent experiments to the problem described above. In particular, we will classify all complete games (a basic class of games) of less than eight players according to they are: a weighted voting game or a game which is (k - 1)-trade robust but not k-trade robust for all values of k. As a consequence it will we showed the existence of games with less than k 2 players which are (k - 1)-trade robust but not k-trade robust. We want to point out that the classifications obtained in this paper by means of experiments are new in the mentioned fields. 1 Introduction Simple games can be viewed as models of voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. Definition 1. A simple game G is a pair (N, W) in which N = {1, 2,...,n} and W is a collection of subsets of N that satisfies: N ∈W, ∅ / ∈W and (mono- tonicity) S ∈W and S ⊆ T ⊆ N then T ∈W. Any set of voters is called a coalition, and the set N is called the grand coali- tion. Members of N are called players or voters, and the subsets of N that are in W are called winning coalitions. The intuition here is that a set S is a winning ⋆ This research was supported by the Spanish “Ministerio de Ciencia y Tecnolog´ ıa” programme TIC2002-00190 (AEDRI II), and Grant BFM 2003–01314 of the Science and Technology Spanish Ministry and the European Regional Development Fund.